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%I #33 Apr 01 2020 10:03:21
%S 1,1,1,4,19,91,476,2586,14421,82225,476913,2804880,16689036,100276894,
%T 607588840,3708251888,22776251835,140676848445,873210347555,
%U 5444307431052,34080036632565,214104150405915,1349504948018208,8531467913710560,54083412667272300,343715994386622918,2189505804590364876
%N Related to enumeration of alkane systems - see reference for precise definition.
%C a(n) is the "number of all staggered conformers of alkyls containing n carbon atoms". It is related to sequence b(n) = A001764(n), which is the number of "space positions of conformers of alkyls related to another alkyl without C_3 symmetry" that contain n carbon atoms. The generating functions of the sequences (a(n): n >= 0) and (b(n): n >= 0), with a(0) = b(0) = 1, appear in some of the papers below. - _Petros Hadjicostas_, Jul 24 2019
%H S. J. Cyvin, <a href="https://doi.org/10.1007/BF01164853">Algebraic solution for the numbers of staggered conformers of alkanes</a>, J. Math. Chem. 17 (1995), 291-293.
%H S. J. Cyvin, J. Brunvoll, B. N. Cyvin, and W. Lüttke, <a href="https://doi.org/10.1515/zna-1995-0911">Enumeration of the staggered conformers of alkanes</a>, Zeitschrift für Naturforschung A 50(9) (1995), 857-863.
%H S. J. Cyvin, B. N. Cyvin, J. Brumvoll, and Jianji Wang, <a href="https://doi.org/10.1016/S0022-2860(97)00419-5">Enumeration of staggered conformers of alkanes and monocyclic cycloalkanes</a>, J. Molec. Struct. 445 (1998), 127-137.
%H S. J. Cyvin, Jianji Wang, J. Brunvoll, Shiming Cao, Ying Li, B. N. Cyvin, and Yugang Wang, <a href="https://doi.org/10.1016/S0022-2860(97)00025-2">Staggered conformers of alkanes: complete solution of the enumeration problem</a>, J. Molec. Struct. 413-414 (1997), 227-239.
%H Jianji Wang, Shiming Cao, and Ying Li, <a href="https://doi.org/10.1007/BF01165165">An algebraic solution for the numbers of staggered conformers of alkanes</a>, J. Math. Chem. 20 (1996), 211-212.
%F From _Petros Hadjicostas_, Jul 24 2019: (Start)
%F We have a(0) = 1, while for n >= 1 we have
%F a(n) = (1/3) * A001764(n) = binomial(3*n, n)/(3*(2*n + 1)) if n !== 1 (mod 3), and
%F a(n) = (1/3) * A001764(n) + (2/3) * A001764((n-1)/3) if n == 1 (mod 3).
%F G.f.: 1 + (x/3) * (B(x)^3 + 2*B(x^3)), where B(x) is the g.f. of sequence A001764, which satisfies the functional equation B(x) = 1 + x*B^3(x). (It also satisfies the equation B(x) = 1/(1 - x*B^2(x)).) We have
%F B(x) = (2/sqrt(3*x)) * sin((1/3) * arcsin(sqrt(27*x/4))).
%F (End)
%p A121179 := proc(n)
%p if n = 0 then
%p return 1;
%p elif modp(n,3) <> 1 then
%p A001764(n) ;
%p else
%p A001764(n)+2*A001764((n-1)/3) ;
%p end if;
%p %/3 ;
%p end proc:
%p seq(A121179(n),n=0..30) ; # _R. J. Mathar_, Jul 31 2019
%t b[n_] := Binomial[3n, n]/(2n + 1);
%t a[n_] := If[n == 0, 1, If[Mod[n, 3] != 1, b[n], b[n] + 2 b[(n-1)/3]]/3];
%t a /@ Range[0, 30] (* _Jean-François Alcover_, Apr 01 2020 *)
%Y Cf. A001764, A121190.
%K nonn
%O 0,4
%A _N. J. A. Sloane_, Aug 15 2006
%E More terms from _Petros Hadjicostas_, Jul 24 2019
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